# The hypergroupoid of boundary conditions for local quantum observables

December 09, 2016

We review the definition of hypergroups by Sunder, and we associate a
hypergroup to a type III subfactor $N\subset M$ of finite index, whose
canonical endomorphism $\gamma\in\mathrm{End}(M)$ is multiplicity-free. It is
realized by positive maps of $M$ that have $N$ as fixed points. If the depth is
$>2$, this hypergroup is different from the hypergroup associated with the
fusion algebra of $M$-$M$ bimodules that was Sunder's original motivation to
introduce hypergroups.
We explain how the present hypergroup, associated with a suitable subfactor,
controls the composition of transparent boundary conditions between two
isomorphic quantum field theories, and that this generalizes to a hypergroupoid
of boundary conditions between different quantum field theories sharing a
common subtheory.

open access link

@article{Bischoff:2016rpu,
author = "Bischoff, Marcel and Rehren, Karl-Henning",
title = "{The hypergroupoid of boundary conditions for local
quantum observables}",
journal = "Adv. Stud. Pure Math.",
volume = "80",
year = "2019",
pages = "32-42",
eprint = "1612.02972",
archivePrefix = "arXiv",
primaryClass = "math-ph",
SLACcitation = "%%CITATION = ARXIV:1612.02972;%%"
}

Keywords:

algebraic quantum field theory, hypergroups, defect, boundary conditions